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The shell10x element term based on the Reissner-Mindlin theory [1], [2],
corresponding to a shell of thickness .

The term requires a custom 3D quadrature, where the components of
quadrature point coordinates are transformed from to
, and the quadrature weights are multiplied by
. The variables and have to use
Shell10XField and
have six components. The reference element mapping is implemented by
Shell10XMapping. The term does not
implement the piezo-electric components of the shell10x element yet.

The term has to be used with quadrilateral cells in 3D and should behave as
the linear elastic term, but with fewer degrees of freedom for the same
accuracy for shell-like structures. The shell has six degrees of freedom in
each of the four nodes: , . The strain and stress
vectors are calculated in a local (co-rotational) coordinate system given
by basis vectors , and
. It holds that

where

is a nodal DOF transformation matrix.

The local displacements , and at any point
in the layer characterized by the isoparametric coordinates ,
and () are interpolated from the nodal
displacement and rotation values (i.e. both membrane and bending
components) using standard isoparametric approximation functions for a
quadrilateral, hence

where , and are
the bending components of displacements calculated from displacements due
to rotations and about
local nodal axes as

where . The local nodal axes
are constructed in order to describe the
behavior of warped (non-planar) elements adequately.